Minimal spanning arborescence

TL;DR

This work initiates a probabilistic study of the minimal spanning arborescence (MSA), the directed analogue of the MST, focusing on infinite-volume limits and geometric structure. It introduces the Chu-Liu-Edmonds-Bock (CLEB) algorithm and the loop contracting random walk (LCRW) as sampling mechanisms, revealing connections to Wilson’s algorithm and invasion percolation. Under broad conditions (transience and nonamenability), the wired MSA limit exists almost surely and decomposes into infinitely many infinite components, typically with one-ended components, while the MSA distribution is shown to depend on the weight distribution. The paper also develops perturbation results, extends the CLEB framework to infinite graphs, and provides unimodular-graph results, complemented by simulations and open questions for future research.

Abstract

We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient trees, the infinite volume limit exists almost surely. We also prove that for nonamenable, unimodular graphs, the limit is almost surely one-ended assuming a certain sufficient condition that guarantees the existence of the limit. This object cannot be studied using well-known algorithms, such as Kruskal's or Prim's algorithm, to sample the minimal spanning tree which has been instrumental in getting analogous results about them (Lyons, Peres, and Schramm). Instead, we use a recursive algorithm due to Chu, Liu, Edmonds, and Bock, which leads to a novel stochastic process which we call the \emph{loop contracting random walk}. This is similar to the well-known and widely studied loop erased random walk, except instead of erasing loops we contract them. The full algorithm bears similarities with the celebrated Wilson's algorithm to generate uniform spanning trees and can be seen as a certain limit of the original Wilson's algorithm.

Figures (9)

• Figure 1: The weights are depicted by the numbers. The ranking of the weights are the same, but have different MSAs as drawn in red
• Figure 2: All the spanning arborescences are marked red.
• Figure 4: The CLEB algorithm. The final output is in the bottom left figure.
• Figure 5: CLEB walk started at $x$.
• Figure 6: Left: The black part is $S_{\tau_{N_1}+1}$, the blue part is $S_{\tau_{N_2}+1} \setminus S_{\tau_{N_1}+1}$ and the violet part is $S_{\tau_{N_3}+1} \setminus S_{\tau_{N_2}+1}$. The sequence of loops contracted is numbered in order. Centre: The path $P_{\tau}$. Colors indicate the color of the edges which are contracted into a single vertex. Right: $E_{N_1}$ is shown in black, $E_{N_2}$ in blue and $E_{N_3}$ in violet. Together, they form $\Gamma_v$

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• proof : Proof that the original CLEB algorithm produces the MSA